Difference between revisions of "User:Timothee Flutre/Notebook/Postdoc/2011/11/10"

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(Bayesian model of univariate linear regression for QTL detection: finish marginal posterior of B)
(Bayesian model of univariate linear regression for QTL detection: add what rest to be done)
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<math>\mathsf{P}(B | Y, X, \tau) \propto exp(B^T (\Sigma_B^{-1} + X^TX) B - Y^TXB -B^TX^TY)</math>
 
<math>\mathsf{P}(B | Y, X, \tau) \propto exp(B^T (\Sigma_B^{-1} + X^TX) B - Y^TXB -B^TX^TY)</math>
  
Let's define <math>\Omega = (\Sigma_B^{-1} + X^TX)^{-1}</math>. We can see that <math>\Omega^T=\Omega</math>, which means that <math>\Omega</math> is a [http://en.wikipedia.org/wiki/Symmetric_matrix symmetric matrix].
+
Importantly, let's define:
 +
 
 +
<math>\Omega = (\Sigma_B^{-1} + X^TX)^{-1}</math>
 +
 
 +
We can see that <math>\Omega^T=\Omega</math>, which means that <math>\Omega</math> is a [http://en.wikipedia.org/wiki/Symmetric_matrix symmetric matrix].
 
This is particularly useful here because we can use the following equality: <math>\Omega^{-1}\Omega^T=I</math>.
 
This is particularly useful here because we can use the following equality: <math>\Omega^{-1}\Omega^T=I</math>.
  
Line 140: Line 144:
 
* '''Joint posterior (2)''': sometimes it is said that the joint posterior follows a Normal Inverse Gamma distribution:
 
* '''Joint posterior (2)''': sometimes it is said that the joint posterior follows a Normal Inverse Gamma distribution:
  
<math>B, \tau | Y, X \sim \mathcal{N}IG(\Omega X^TY, \tau^{-1}\Omega, \frac{N+\kappa}{2}, \frac{\lambda^\ast}{2})</math>
+
<math>B, \tau | Y, X \sim \mathcal{N}IG(\Omega X^TY, \; \tau^{-1}\Omega, \; \frac{N+\kappa}{2}, \; \frac{\lambda^\ast}{2})</math>
  
 
where <math>\lambda^\ast = (Y^T X \Omega X^T Y + Y^T Y + \lambda)</math>
 
where <math>\lambda^\ast = (Y^T X \Omega X^T Y + Y^T Y + \lambda)</math>
Line 155: Line 159:
 
<math>\mathsf{P}(B | Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}} \Gamma(\frac{N+\kappa+3}{2})}{(2\pi)^\frac{3}{2} |\Omega|^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \left( \frac{\lambda^\ast}{2} + (B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY) \right)^{-\frac{N+\kappa+3}{2}}</math>
 
<math>\mathsf{P}(B | Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}} \Gamma(\frac{N+\kappa+3}{2})}{(2\pi)^\frac{3}{2} |\Omega|^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \left( \frac{\lambda^\ast}{2} + (B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY) \right)^{-\frac{N+\kappa+3}{2}}</math>
  
And we now recognize a [http://en.wikipedia.org/wiki/Multivariate_t-distribution multivariate t distribution]:
+
And we now recognize a [http://en.wikipedia.org/wiki/Multivariate_t-distribution multivariate Student's t-distribution]:
  
 
<math>\mathsf{P}(B | Y, X) = \frac{\Gamma(\frac{N+\kappa+3}{2})}{\Gamma(\frac{N+\kappa}{2}) \pi^\frac{3}{2} |\lambda^\ast \Omega|^{\frac{1}{2}} } \left( 1 + \frac{(B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY)}{\lambda^\ast} \right)^{-\frac{N+\kappa+3}{2}}</math>
 
<math>\mathsf{P}(B | Y, X) = \frac{\Gamma(\frac{N+\kappa+3}{2})}{\Gamma(\frac{N+\kappa}{2}) \pi^\frac{3}{2} |\lambda^\ast \Omega|^{\frac{1}{2}} } \left( 1 + \frac{(B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY)}{\lambda^\ast} \right)^{-\frac{N+\kappa+3}{2}}</math>
 +
 +
We hence can write:
 +
 +
<math>B | Y, X \sim \mathcal{S}_{N+\kappa}(\Omega X^TY, \; (Y^T X \Omega X^T Y + Y^T Y + \lambda) \Omega)</math>
 +
 +
 +
* '''Bayes Factor''': to do
 +
 +
 +
* '''In practice''': to do
 +
 +
invariance properties motivate the use of limits for some "unimportant" hyperparameters
 +
 +
average BF over grid
 +
 +
 +
* '''R code''': to do
  
 
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<!-- ##### DO NOT edit below this line unless you know what you are doing. ##### -->

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Bayesian model of univariate linear regression for QTL detection

See Servin & Stephens (PLoS Genetics, 2007).


  • Data: let's assume that we obtained data from N individuals. We note [math]y_1,\ldots,y_N[/math] the (quantitative) phenotypes (e.g. expression levels at a given gene), and [math]g_1,\ldots,g_N[/math] the genotypes at a given SNP (encoded as allele dose: 0, 1 or 2).


  • Goal: we want to assess the evidence in the data for an effect of the genotype on the phenotype.


  • Assumptions: the relationship between genotype and phenotype is linear; the individuals are not genetically related; there is no hidden confounding factors in the phenotypes.


  • Likelihood: we start by writing the usual linear regression for one individual

[math]\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i \text{ with } \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})[/math]

where [math]\beta_1[/math] is in fact the additive effect of the SNP, noted [math]a[/math] from now on, and [math]\beta_2[/math] is the dominance effect of the SNP, [math]d = a k[/math].

Let's now write the model in matrix notation:

[math]Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T[/math]

This gives the following multivariate Normal distribution for the phenotypes:

[math]Y | X, \tau, B \sim \mathcal{N}(XB, \tau^{-1} I_N)[/math]

Even though we can write the likelihood as a multivariate Normal, I still keep the term "univariate" in the title because the covariance matrix of [math]Y | X, \tau, B[/math] remains a single real number, [math]\tau[/math].

The likelihood of the parameters given the data is therefore:

[math]\mathcal{L}(\tau, B) = \mathsf{P}(Y | X, \tau, B)[/math]

[math]\mathcal{L}(\tau, B) = \left(\frac{\tau}{2 \pi}\right)^{\frac{N}{2}} exp \left( -\frac{\tau}{2} (Y - XB)^T (Y - XB) \right)[/math]


[math]\mathsf{P}(\tau, B) = \mathsf{P}(\tau) \mathsf{P}(B | \tau)[/math]

A Gamma distribution for [math]\tau[/math]:

[math]\tau \sim \Gamma(\kappa/2, \, \lambda/2)[/math]

which means:

[math]\mathsf{P}(\tau) = \frac{\frac{\lambda}{2}^{\kappa/2}}{\Gamma(\frac{\kappa}{2})} \tau^{\frac{\kappa}{2}-1} e^{-\frac{\lambda}{2} \tau}[/math]

And a multivariate Normal distribution for [math]B[/math]:

[math]B | \tau \sim \mathcal{N}(\vec{0}, \, \tau^{-1} \Sigma_B) \text{ with } \Sigma_B = diag(\sigma_{\mu}^2, \sigma_a^2, \sigma_d^2)[/math]

which means:

[math]\mathsf{P}(B | \tau) = \left(\frac{\tau}{2 \pi}\right)^{\frac{3}{2}} |\Sigma_B|^{-\frac{1}{2}} exp \left(-\frac{\tau}{2} B^T \Sigma_B^{-1} B \right)[/math]


  • Joint posterior (1):

[math]\mathsf{P}(\tau, B | Y, X) = \mathsf{P}(\tau | Y, X) \mathsf{P}(B | Y, X, \tau)[/math]


  • Conditional posterior of B:

[math]\mathsf{P}(B | Y, X, \tau) = \mathsf{P}(B, Y | X, \tau)[/math]

[math]\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B, Y | X, \tau)}{\mathsf{P}(Y | X, \tau)}[/math]

[math]\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B | \tau) \mathsf{P}(Y | X, B, \tau)}{\int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B}[/math]

Here and in the following, we neglect all constants (e.g. normalization constant, [math]Y^TY[/math], etc):

[math]\mathsf{P}(B | Y, X, \tau) \propto \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B)[/math]

We use the prior and likelihood and keep only the terms in [math]B[/math]:

[math]\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B) exp((Y-XB)^T(Y-XB))[/math]

We expand:

[math]\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B - Y^TXB -B^TX^TY + B^TX^TXB)[/math]

We factorize some terms:

[math]\mathsf{P}(B | Y, X, \tau) \propto exp(B^T (\Sigma_B^{-1} + X^TX) B - Y^TXB -B^TX^TY)[/math]

Importantly, let's define:

[math]\Omega = (\Sigma_B^{-1} + X^TX)^{-1}[/math]

We can see that [math]\Omega^T=\Omega[/math], which means that [math]\Omega[/math] is a symmetric matrix. This is particularly useful here because we can use the following equality: [math]\Omega^{-1}\Omega^T=I[/math].

[math]\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Omega^{-1} B - (X^TY)^T\Omega^{-1}\Omega^TB -B^T\Omega^{-1}\Omega^TX^TY)[/math]

This now becomes easy to factorizes totally:

[math]\mathsf{P}(B | Y, X, \tau) \propto exp((B^T - \Omega X^TY)^T\Omega^{-1}(B - \Omega X^TY))[/math]

We recognize the kernel of a Normal distribution, allowing us to write the conditional posterior as:

[math]B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)[/math]


  • Posterior of [math]\tau[/math]:

Similarly to the equations above:

[math]\mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \mathsf{P}(Y | X, \tau)[/math]

But now, to handle the second term, we need to integrate over [math]B[/math], thus effectively taking into account the uncertainty in [math]B[/math]:

[math]\mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B[/math]

Again, we use the priors and likelihoods specified above (but everything inside the integral is kept inside it, even if it doesn't depend on [math]B[/math]!):

[math]\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} \tau^{N/2} exp(-\frac{\tau}{2} B^T \Sigma_B^{-1} B) exp(-\frac{\tau}{2} (Y - XB)^T (Y - XB)) \mathsf{d}B[/math]

As we used a conjugate prior for [math]\tau[/math], we know that we expect a Gamma distribution for the posterior. Therefore, we can take [math]\tau^{N/2}[/math] out of the integral and start guessing what looks like a Gamma distribution. We also factorize inside the exponential:

[math]\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} exp \left[-\frac{\tau}{2} \left( (B - \Omega X^T Y)^T \Omega^{-1} (B - \Omega X^T Y) - Y^T X \Omega X^T Y + Y^T Y \right) \right] \mathsf{d}B[/math]

We recognize the conditional posterior of [math]B[/math]. This allows us to use the fact that the pdf of the Normal distribution integrates to one:

[math]\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} exp\left[-\frac{\tau}{2} (Y^T X \Omega X^T Y + Y^T Y) \right][/math]

We finally recognize a Gamma distribution, allowing us to write the posterior as:

[math]\tau | Y, X \sim \Gamma \left( \frac{N+\kappa}{2}, \; \frac{1}{2} (Y^T X \Omega X^T Y + Y^T Y + \lambda) \right)[/math]


  • Joint posterior (2): sometimes it is said that the joint posterior follows a Normal Inverse Gamma distribution:

[math]B, \tau | Y, X \sim \mathcal{N}IG(\Omega X^TY, \; \tau^{-1}\Omega, \; \frac{N+\kappa}{2}, \; \frac{\lambda^\ast}{2})[/math]

where [math]\lambda^\ast = (Y^T X \Omega X^T Y + Y^T Y + \lambda)[/math]


  • Marginal posterior of B: we can now integrate out [math]\tau[/math]:

[math]\mathsf{P}(B | Y, X) = \int \mathsf{P}(B, \tau | Y, X) \mathsf{d}\tau[/math]

[math]\mathsf{P}(B | Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}}}{(2\pi)^\frac{3}{2} |\Omega|^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \int \tau^{\frac{N+\kappa+3}{2}-1} exp \left[-\tau \left( \frac{\lambda^\ast}{2} + (B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY) \right) \right] \mathsf{d}\tau[/math]

Here we recognize the formula to integrate the Gamma function:

[math]\mathsf{P}(B | Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}} \Gamma(\frac{N+\kappa+3}{2})}{(2\pi)^\frac{3}{2} |\Omega|^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \left( \frac{\lambda^\ast}{2} + (B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY) \right)^{-\frac{N+\kappa+3}{2}}[/math]

And we now recognize a multivariate Student's t-distribution:

[math]\mathsf{P}(B | Y, X) = \frac{\Gamma(\frac{N+\kappa+3}{2})}{\Gamma(\frac{N+\kappa}{2}) \pi^\frac{3}{2} |\lambda^\ast \Omega|^{\frac{1}{2}} } \left( 1 + \frac{(B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY)}{\lambda^\ast} \right)^{-\frac{N+\kappa+3}{2}}[/math]

We hence can write:

[math]B | Y, X \sim \mathcal{S}_{N+\kappa}(\Omega X^TY, \; (Y^T X \Omega X^T Y + Y^T Y + \lambda) \Omega)[/math]


  • Bayes Factor: to do


  • In practice: to do

invariance properties motivate the use of limits for some "unimportant" hyperparameters

average BF over grid


  • R code: to do